Energy-Efficient Time-Stampless Adaptive Nonuniform Sampling

ABSTRACT

Described herein is a sampling system and related sampling scheme. The system and sampling scheme is based upon a framework for adaptive non-uniform sampling schemes. In the system and schemes described herein, time intervals between samples can be computed by using a function of previously taken samples. Therefore, keeping sampling times (time-stamps), except initialization times, is not necessary. One aim of this sampling framework is to provide a balance between reconstruction distortion and average sampling rate. The function by which sampling time intervals can be computed is called the sampling function. The sampling scheme described herein can be applied appropriately on different signal models such as deterministic or stochastic, and continuous or discrete signals. For each different signal model, sampling functions can be derived.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.61/504,440, which is hereby incorporated by reference in its entirety.

GOVERNMENT RIGHTS

This invention was made with government support under Contract No.FA9550-09-1-0196 awarded by the Air Force Office of Scientific Research.The government has certain rights in this invention.

FIELD OF THE INVENTION

The concepts described herein relate to sampling and more particularlyto adaptive nonuniform sampling schemes.

BACKGROUND OF THE INVENTION

As is known in the art, there is an increasing trend for mobile phones,“smart” phones, sensors and other lightweight and battery-operateddevices to increasingly affect the everyday life of human's. Suchdevices have already started to become more and more intelligent,equipped with a wide range of features.

The majority of these devices, however, are resource constrained. Forexample, the devices are typically powered by batteries, or even byenergy scavenging, and thus the devices typically have strict powerbudgets. Thus, reducing power consumption of such devices is beneficialfor battery lifetime.

One problem, however, is that many applications require such devices tocontinuously measure different quantities such as voice, acceleration,light or voltage levels. After processing these quantities andextracting required features, the devices then typically communicateinformation to humans or to other devices. A concern in all of theseapplications is the large amount of power which is consumed tocontinuously sample, process and transmit information.

As is also known, a significant challenge is posed by sampling a signalwhile at the same time trying to satisfy sampling rate andreconstruction error requirements. This is particularly true in certainapplications within the signal processing domain. It should beappreciated that in many applications, energy consumed during a samplingprocess can be a significant part of a system's energy consumption.

A variety of sampling techniques are known. Sampling techniques whichfollow a Nyquist sampling theorem utilize an appropriate uniformsampling setup for band-limited deterministic signals. One problem withsuch an approach, however, is that some samples may be redundant becausethe maximum bandwidth of the signal may not be a good measure of signalvariations at different times. Redundant samples result in extra powerconsumption in the sampling procedure as well as in processes whichfollow the sampling. For example, if it is necessary, to transmitsamples to another location, having a relatively large number ofadditional samples results in higher transmission energy and/or in extraenergy spent for compressing the samples.

As a way to more efficiently sample a signal, adaptive nonuniformsampling schemes have been proposed. Such adaptive nonuniform samplingschemes result in reduced power consumption since the number ofredundant samples is decreased or, in some cases, even eliminated.

Several non-uniform adaptive sampling schemes are known. For instance, anon-uniform sampling scheme based upon level-crossings with iterativedecoding has been proposed as has an approach based upon level crossingswith a filtering technique which adapts the sampling rate and filterorder by analyzing the input signal variations. Also, two adaptivesampling schemes for band-limited deterministic signals have also beenproposed. One scheme is based upon linear time-varying low pass filtersand another scheme is based upon time-warping of band-limited signals.

Such non-uniform sampling schemes give rise to at least two problemswhich make it difficult to apply these schemes in practicalapplications. First, non-uniform sampling schemes are designed forspecific signal models (i.e., they are not generic). This is because itis difficult to determine the next sampling time step at each time (i.e.rate control). Second, it is necessary to keep (e.g. store) or transmitsampling times since they are required in the reconstruction process.

For a discrete stochastic signal, one sampling scheme samples uniformlyat a fixed high sampling rate (e.g. using an analog to digital converter(ADC)). Source coding is then used to compress these samplesapproximately to their entropy rate before transmission. While thistechnique is theoretically optimal, in practice this technique has someinefficiencies in terms of extra sampling and processing power required.Moreover, to achieve desired performance levels, long blocks of samplesare necessary to be able to use source coding efficiently. This isparticularly true if statistical properties of the signal vary slowly intime. This block-based approach may lead to a large delay on thereconstruction side.

It would therefore, be desirable, to provide an adaptive, nonuniformsampling technique which is generic (i.e. can be used in a wide varietyof applications) and which does not need to store or transmit samplingtimes.

SUMMARY OF THE INVENTION

It has, in accordance with the concepts, systems and techniquesdescribed herein, been recognized that a smart sampling scheme shouldtake samples as needed for a particular application (i.e., the techniqueshould utilize innovative samples). This approach leads to an adaptivesampling scheme which inherently is nonuniform.

With this understanding, a general technique for efficiently samplingsignals in an adaptive way is provided. The technique reduces (or insome cases, even minimizes) a required sampling rate and therefore theenergy consumed during a sampling process. Thus, the technique describedherein can greatly improve the energy consumption of a device.

The adaptive nonuniform sampling framework described herein has twocharacteristics. First, the adaptive nonuniform sampling framework takessamples only when they are innovative for the considered application.This reduces the number of required measurements. Second, unliketraditional nonuniform sampling procedures, sampling times need not betransmitted since the receiver can recover them within the context ofthe framework. This saves transmission rate and power. This technique(or sampling framework) is referred to herein as Time-Stampless AdaptiveNonuniform Sampling (TANS) or Energy-Efficient TANS (ETANS). TANS can beused in a wide variety of applications including but not limited tomodem systems to provide significant rate and power benefits.

One key idea of the TANS technique described herein is that timeintervals between samples can be computed by using a function ofpreviously taken samples. Therefore, keeping sampling times(time-stamps), except initialization times, is not necessary. The aim ofthis sampling framework is to have a balance between the reconstructiondistortion and the average sampling rate. The function by which samplingtime intervals can be computed is referred to as a sampling function.

This sampling structure can be applied appropriately on different signalmodels such as deterministic or stochastic, and continuous or discretesignals, and for each, a different sampling function can be derived.

In contrast to schemes which take samples of discrete stochastic signalsuniformly at a fixed high rate using an ADC, the techniques describedherein takes samples when they are innovative. Since the power consumedduring the sampling process depends linearly on the sampling frequency,the time-stampless, adaptive, nonuniform sampling techniques describedherein are more efficient in terms of power and processing time, thanprior art techniques which uniformly sample at a fixed high rate usingan ADC.

Furthermore, in contrast to block-based approaches which may lead to alarge delay on the reconstruction side, the techniques described hereincorrespond to substantially real-time delay-free compression schemes.These schemes adaptively compress the signal by using local propertiesof the signal causally which results in reduced power consumption.

In one embodiment, a sampling family which can be applied on bothdeterministic and stochastic signals, satisfying sampling andreconstruction requirements is described. This sampling family is aclass of locally adaptive sampling (LAS) schemes. In this samplingfamily, time intervals between samples are computed via a samplingfunction which is based upon previously taken samples. Hence, althoughit is a non-uniform sampling scheme, it is not necessary to keepsampling times. Thus, in accordance with the present concepts, systemand techniques described herein, described herein is a real-timecompression scheme which adaptively compresses a signal by using itslocal properties causally.

In one embodiment, the aim of LAS is to have the average sampling rateand the reconstruction error satisfy some requirements. Four differentschemes of LAS are described herein with two schemes being designed fordeterministic signals and two schemes being designed for stochasticsignals.

In the schemes designed for deterministic signals, a Taylor SeriesExpansion (TSE) sampling function is derived. In one embodiment, the TSEassumes only that the third derivative of the signal is bounded, butrequires no other specific knowledge of the signal. Then, a DiscreteTime-Valued (DTV) sampling function is proposed, where the sampling timeintervals are chosen from a lattice.

In the schemes designed for stochastic signals, two sampling methodsbased upon linear prediction filters are described. A first samplingmethod is a Generalized Linear Prediction (GLP) sampling function, and asecond sampling function is a Linear Prediction sampling function withSide Information (LPSI). In the GLP method, one only assumes the signalis locally stationary. However, LPSI is specifically designed for aknown signal model.

With this particular arrangement, an adaptive, nonuniform samplingframework which can be used on both deterministic and stochastic signalsis provided. By taking samples only when they are innovative for theconsidered application, the number of required measurements is reduced.Also, since the receiver can recover the sample times, and a continuousstream of sampling times need not be transmitted. This savestransmission rate and power.

It should be noted that in conventional nonuniform sampling procedures,sampling times must be transmitted to the receiver which impacts thesystem transmission rate and also utilizes power.

It should be appreciated that sampling functions can use either fixed orvariable window sizes. Alternatively still, a combination of both fixedand variable window sizes can be used. In systems which utilize avariable window size, the system dynamically learns to adjust the window(e.g. the number of samples) for a particular signal. In a fixed windowsystem, the number of samples to use in a window is selected based uponthe needs of the particular application. For applications which requirea high degree of accuracy, a relatively large number of samples in eachwindow is required (i.e. need a large M). Some factors to consider inselecting number of samples to use in a window include buffer size sincethe larger the value of M, the larger the buffer which will be needed.It should also be appreciated that the higher the value of M, the morecomplicated the implementation. Thus, a trade-off must be made betweenthe number of samples needed, the power consumption, and the ability topredict the future of the next window with a desired level of accuracy.

For stochastic signals the sampling function prediction can be basedupon a plurality of different methods such as a greedy Method or aDynamic Programming Based Method. For deterministic signals, a TaylorSeries Expansion or a Discrete-Time Valued Method can be used. Inoperation, the future of the signals is predicted, then depending uponthe allowed distortion (or other error value or metric), the next sampleis taken.

Thus, the general framework described herein provides at least twoadvantages: (1) the system is adaptive—i.e. it is possible to changetime intervals based upon variations (ideally this is inherently in thedesign of the sampling function which is chosen); and (2) the system istime-stamp-less—i.e. there is no need to store or transmit samplingtimes after the first M sampling times because those times after thefirst M sampling times can be computed using the sampling function.

Schemes of TANS may be categorized into parametric and non-parametricmethods. In parametric methods, the signal model is known a priori andthe sampling function is designed for that model. In non-parametricschemes, the signal model is not known (or is known partially) and thesampling function learns the necessary characteristics of the signal byusing the taken samples.

It should be appreciated that in the case of stochastic signals, thereare primarily two general techniques for designing a sampling function:(1) a greedy sampling Function; (2) dynamic programming based methods;(or approximate dynamic programming based methods).

In the greedy approach, the system takes the next sample to minimize thesum of reconstruction costs and a sampling rate penalty.

Dynamic programming based methods are similar to the greedy method, butthey also consider quality (Q) of the next sampling prediction or state.Thus, in the DP approach, the concept is trade-off between a presentstate and a future state (e.g. allow lower performance in a presentstate to improve a future state).

In accordance with one aspect of the concepts described herein, a methodof designing a sampling function includes (a) obtaining an initialnumber of samples; (b) predicting a future sample upon the initialnumber of samples; (c) computing an error value of the predicted futuresample; (d) comparing the error value to a threshold value (e.g. alloweddistortion); (e) in response to the error value exceeding the thresholdvalue, taking a sample; and (f) in response to the error value notexceed the threshold value increasing the sampling time and computing anew predicted future sample.

With this particular arrangement, a technique for adaptive,time-stamp-less sampling is provided. The technique is adaptive since itis possible to change time intervals based upon signal variations. Insome embodiments, this characteristic is a result of a design of aselected sampling function. The technique is time-stamp-less since thereis no need to store or transmit sampling times after the first Msampling times because those times after the first M sampling times canbe computed using the sampling function.

In accordance with a further aspect of the concepts described herein, amethod of sampling includes (a) obtaining samples of a signal; and (b)computing a next sample time only as a function of the previously takensamples.

With this particular arrangement, an adaptive and time-stamplesssampling technique is provided. In one embodiment, computing a nextsample time as a function of previously taken samples is performed bycomputing a next sample time as a function of only a predeterminednumber of previously taken samples (e.g. M previously taken samples).

In accordance with a still further aspect of the concepts describedherein, a sampling method includes (a) obtaining M samples of a signal;and (b) computing a next sample time as a function of only M previouslytaken samples of the signal wherein the sampling is adaptive andself-synchronizing. The method further includes (c) recovering samplingtimes using only the sample values and the first M sample times taken ofthe signal.

In accordance with yet a further aspect of the concepts, systems, andtechniques described herein, a sampling system and method in which timeintervals between samples are computed as a function of previously takensamples is described. With this technique, sample intervals are computedvia a sampling function corresponding to a time-stampless adaptivenonuniform sampling technique.

In accordance with a further aspect of the concepts, system andtechniques described herein, a locally adaptive sampling (LAS) systemand method designed for deterministic signals is described. In oneembodiment, the system and method utilize a Taylor series expansion(TSE) sampling function. In one embodiment, the TSE only assumes thethird derivative of the signal is banded (but requires no other specificknowledge of the signal)

In accordance with a still further aspect of the concepts, systems andtechniques described herein, locally adaptive sampling (LAS) system andmethod for deterministic signals which utilizes a discrete time-valued(DIV) sampling function is described. In one embodiment, the LAS systemand method utilize sampling time intervals are chosen from a lattice.

In accordance with a still further aspect of the concepts, systems andtechniques a locally adaptive sampling (LAS) system and method forstochastic signals comprise a sampling method based upon a linearprediction filter which utilizes a linear prediction (LP) samplingfunction with side information (LPSI). This technique is specificallydesigned for a known signal model.

In one embodiment, a locally adaptive sampling (LAS) system and methodutilizes sampling time intervals which are discrete values.

In one embodiment, a locally adaptive sampling (LAS) system and methodfor discrete stochastic signals utilizes a sampling function based upona generalized linear prediction (GAP) filter. In one embodiment, thistechnique assumes that the signal is locally stationary.

In accordance with a still further aspect of the concepts, systems andtechniques, a method for providing an appropriate sampling function fora time-stampless adaptive nonuniform sampling (TANS) system and methodis described. Factors to consider in providing a sampling functioninclude, but are not limited to: the particular application (e.g. thecharacteristics of the signals involved), the signal model, thereconstruction method and sampling requirements (e.g. sampling rate,distortion requirements, etc.).

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of this invention, as well as the inventionitself, may be more fully understood from the following description ofthe drawings in which:

FIG. 1 is a plot of an amplitude of a continuous signal vs. time;

FIG. 2 is a block diagram of a system for sampling and transmittingdata;

FIG. 3 is a flow diagram of a fixed window process for time-stamplessadaptive nonuniform sampling (TANS);

FIG. 4 is a flow diagram of a process for time-stampless adaptivenonuniform sampling;

FIG. 5 is a flow diagram of a process for time-stampless adaptivenonuniform sampling;

FIG. 6 is a flow diagram of a process for designing a sampling function;

FIG. 7 is a plot of signal amplitude vs. time;

FIG. 8 is a plot of an EKG signal;

FIG. 8A is a plot of a reconstructed EKG signal using TANS; and

FIG. 8B is a plot of a reconstructed EKG signal using uniform sampling.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Before describing a system and process for a process for time-stamplessadaptive nonuniform sampling (TANS), some introductory concepts andterminology are explained.

A deterministic signal is a signal which can be described by analyticalexpressions for all times (past, present, future). Thus, deterministicsignals are predictable for arbitrary times and can be reproducedidentically arbitrarily often.

A stochastic signal may be an analog or continuous parameter signals ormay be made up of a plurality of discrete values (a so-called discretestochastic signal) which is often represented as X(n) where n takes on aplurality of discrete values.

It should be appreciated that the systems and processes described hereinfor time-stampless adaptive nonuniform sampling can be applied to eitherdeterministic or stochastic signals which are either continuous ordiscrete. Accordingly reference to or exemplary embodiments describedherein which use continuous parameter signals should not be construed aslimiting. Rather such examples are provided only to promote clarity inthe text and to promote clarity of the broad concepts described hereinrelated to the adaptive, nonuniform sampling framework.

Reference is sometimes made herein to specific applications. Forexample, energy efficient sampling and communication systems and healthcare/monitoring systems. It should be appreciated that the systems andprocesses described herein for time-stampless adaptive nonuniformsampling can be applied to a wide variety of different applications. Ina broad sense, sampling can be considered as a query for information.The systems and techniques described herein can be used in anyapplications where it is necessary to reconstruct some function f(t)that can only be measured remotely. The systems and techniques describedherein allow one to obtain a sample of the function f(t) at an arbitrarytime when it is queried (e.g. queried over some communication medium).In such an application, a portion of the operational cost (e.g. power)is proportional to the number of samples which are obtained. It shouldbe appreciated that in some applications, this part of the operationalcost is dominant. Thus, in such applications, the systems and techniquesdescribed herein result in large operational cost benefits (e.g. lesspower consumption).

Referring now to FIG. 1, a time varying continuous signal 10 has aplurality of samples 12 a-12 h taken therefrom. A first plurality ofsamples, here samples 12 a-12 f, are taken inside a first window 14. Ingeneral, the first M samples are taken in the first window 14 at timest_(i−m+1) through t_(i). In the particular example of FIG. 1, M=6. Thoseof ordinary skill in the art will appreciate, however, that fewer orless than six (6) samples may be used. The particular number of samplesM to use in any particular application depends upon a variety offactors.

FIG. 1 also illustrates a second window 16 which is the same size asfirst window 14 in terms of the number of samples. Thus, in thisexemplary embodiment, window 16 also includes six (6) samplescorresponding to samples 12 b-12 g. Thus, in this exemplary embodiment afixed window size is used. It should, however, be appreciated that inother embodiments a variable window size may be used (i.e. the windowsneed not contain the same number of samples).

In operation, the future of the signals is predicted, then dependingupon the allowed distortion (or other error value or metric), the nextsample is taken.

With the above arrangement, T_(i) may be expressed as

T _(i) =f(X(t _(i)), . . . T _(i+1))=f(X(T _(i+1)), . . . X(T _(i−m+2)),t _(i+1) , . . . t _(i−m+2))

In this example, a sampling function with order M may be represented asf(.). It should be noted the sampling function is a function ofpreviously taken samples as opposed to being a function of the entiresignal. By not depending upon the entire signal, if the signal has a lowsignal variation, then the sampling time is increased (i.e. the timebetween samples is increased). On the other hand, if the signal has ahigh signal variation, then the sample time is decreased (i.e. the timebetween samples is decreased). It should also be noted that it is notnecessary to keep sampling times.

Designing appropriate sampling functions depends upon a variety offactors including, but not limited to the application, the signal model,the reconstruction method, other reconstruction requirements, thesampling rate, the allowed distortion level, etc. . . . .

For example, in a medical application such as a continuous healthmonitoring application, such as monitoring an EKG signal, the EKG signalis known to contain both low and high frequency components. In thiscase, a generic sampling function (as opposed to an application specificsampling function) can be used. In one embodiment, a generic samplingfunction can be derived by using a Fourier Transform and theCauchy-Schwarz inequality:

$T_{i} = \left( {\frac{p}{c}\frac{1}{{w\left( t_{i} \right)}^{2}}} \right)^{1\text{/}3}$

where

${w({ti})} = {\frac{X_{(t_{i})} - X_{(t_{i - 1})}}{t_{i} - t_{i - 1}}}$

where

w(t_(i)) is an approximation of the signal slope at time t_(i);

c is a sampling parameter; and

p is the signal power.

It should be noted that the higher the derivative at time t_(i), thehigher the signal variations at this time and thus, the smaller thesampling step size.

An example of a sampling function with discrete steps is

$T_{i} = \begin{Bmatrix}{{{T_{i - 1} +} \in {{w({ti})} < {th}}},} \\{{T_{i - 1}{th}},{\leq {w({ti})} \leq {th}_{2}}} \\{{T_{i - 1} -} \in^{\prime}{{w({ti})} \geq {th}_{2}}}\end{Bmatrix}$

in which

${w({ti})} = \frac{{\frac{X_{ti} - t_{i - 1}}{T_{i - 1}} - \frac{X_{({{ti} - 1})} - X_{({{ti} - 2})}}{T_{i - 2}}}}{\left( {T_{i - 1} + T_{i - 2}} \right)/2}$

As noted above, the sampling structure described above can be used withboth deterministic and stochastic modeling and analysis. For example,Taylor Series Expansions and Discrete-Time Valued methods can be usedfor deterministic signals while for stochastic signals the predictioncan also be based upon a plurality of different methods such as a greedyMethod, a Dynamic Programming Based method or a Generalized LinearPrediction method.

It should be appreciated that while the same number of samples are usedin windows 14, 16 (i.e. fixed window sizes), variable window sizes canalso be used.

In a fixed window system, a number of samples to use in a window areselected based upon the needs of the particular application. Forapplications which require a high degree of accuracy, a relatively largenumber of samples in each window is required (i.e. need a large M). Somefactors to consider in selecting number of samples to use in a windowinclude buffer size since the larger the value of M, the larger thebuffer which will be needed. It should also be appreciated that thehigher the value of M, the more complicated the implementation. Thus, atrade-off must be made between the number of samples needed, the powerconsumption, and the ability to predict the future of the next windowwith a desired level of accuracy.

In systems which utilize a variable window size, the system dynamicallylearns how to adjust the window (e.g. the number of samples) for aparticular signal.

It should also be appreciated that a combination of both fixed andvariable window sizes can be used and that some systems may dynamicallychange between fixed and variable window sizes.

Thus, the general framework described herein provides at least twoadvantages: (1) the sampling system is adaptive—i.e. it is possible tochange time intervals based upon variations (ideally this is inherentlyin the design of the sampling function which is chosen); and (2) thesampling system is time-stamp-less—i.e. there is no need to store ortransmit sampling times after the first M sampling times because thosetimes after the first M sampling times can be computed using thesampling function.

As mentioned above, in the case of stochastic signals, there areprimarily two general techniques for designing a sampling function: (1)a greedy Sampling Function; (2) Dynamic Programming Based Methods; (orapproximate dynamic programming based methods). In the greedy approachthe system takes the next sample to minimize the sum of reconstructioncosts and a sampling rate penalty. Dynamic programming based methods aresimilar to the greedy method, but they also consider quality (Q) of thenext sampling prediction or state. Thus, in the DP approach, the conceptis trade-off between a present state and a future state (e.g. allowlower performance in a present state to improve a future state).

Referring now to FIG. 2, a sampling and transmission system includes aprocessing device 30 which comprises an analog to digital converter(ADC) 32 which samples or converts signals 33 provided thereto into astream of bits which may be grouped as digital words. Signal 33 maycorrespond to either deterministic or stochastic signals which areeither continuous (analog) or discrete.

ADC 32 operates in accordance with a sampling function which defines atime-stampless, arbitrary and nonuniform sampling rate. The particularsampling function selected for any particular application is selectedbased upon a variety of factors related to the requirements of theparticular application including, but not limited to factors such as theparticular application, the signal model, the reconstruction method,other reconstruction requirements, the sampling rate and the alloweddistortion level.

At the output of ADC 32, the stream of digital words are coupled to atransmission device 34 and also to a control logic element 36. Controllogic element 16 may include a buffer memory element (not shown in FIG.2) and may be under control of a system controller (not shown in FIG. 2)via control signals on control lines (not shown in FIG. 2). The samplingrate of ADC 32 is under control of control logic element 36 via controlsignals on control lines 38 (i.e. control logic element 36 implementsthe sampling function). After a sufficient amount of data has beencollected in control logic element 16, the control logic element 16computes a time at which a sample should be taken by ADC 12.

Significantly, the sampling function is a function of previously takensamples (as opposed to being a function of the entire signal).Furthermore, the sampling function causes the system 30 to take samplesonly when they are innovative for the particular application (e.g. forthe signal being sampled). This results in fewer samples being taken.Since the power consumed by system 30 is largely dependent upon samplingfrequency, the technique results in system 30 being more efficient interms of power and processing time, than prior art techniques.

Also, since the time intervals between samples are computed using afunction of previously taken samples, it is not necessary to keepsampling times. This results in further power efficiencies since less itis not necessary to transmit sampling times via transmission device 34.

FIGS. 3-6 are a series of flow diagrams showing exemplary processes toperform time-stampless adaptive nonuniform sampling (TANS). Such aprocess may be performed, for example, by a processing apparatus whichmay, for example, be provided as part of a sampling system such as thatshown in FIG. 2. The rectangular elements (e.g. block 40 in FIG. 3) inthe flow diagram(s) are herein denoted “processing blocks” and representsteps or instructions or groups of instructions. Some of the processingblocks can represent an empirical procedure or a database while otherscan represent computer software instructions or groups of instructions.The diamond shaped elements in the flow diagrams (e.g. block 50 in FIG.3) are herein denoted “decision blocks” and represent steps ofinstructions or groups of instructions which affect the processing ofthe processing blocks. It should be noted that some of the processesdescribed in the flow diagram may be implemented via computer softwarewhile others may be implemented in a different manner e.g. via anempirical procedure.

Alternatively, some of the processing blocks can represent stepsperformed by functionally equivalent circuits such as a digital signalprocessor circuit or an application specific integrated circuit (ASIC).The flow diagram does not depict the syntax of any particularprogramming language. Rather, the flow diagram illustrates thefunctional information one of ordinary skill in the art requires toperform the steps or to fabricate circuits or to generate computersoftware to perform the processing required of the particular apparatus.It should be noted that where computer software can be used, manyroutine program elements, such as initialization of loops and variablesand the use of temporary variables are not shown. It will be appreciatedby those of ordinary skill in the art that unless otherwise indicatedherein, the particular sequence of steps described is illustrative onlyand can be varied without departing from the spirit of the concepts,systems and techniques described herein.

Referring now to FIG. 3, an exemplary time-stampless adaptive nonuniformsampling (TANS) process for a fixed window system begins in processingblock 40 by selecting a number of samples to use in each window (hencethe windows are said to be “fixed”). Processing then proceeds toprocessing block 42 in which at least a first M samples within the firstwindow (e.g. window 14 in FIG. 1) are taken.

Next, as shown in processing block 44, a sampling function is used tocompute the next sampling time. This means that a sample should be takenafter a specific amount of time (i.e. there is no need to keep checking;it should be noted that FIG. 4 shows a variation of the process in whichthe process keeps checking to see when it is time to sample). In oneembodiment, processing block 44 may be accomplished as described belowin conjunction with processing blocks 82, 84 of FIG. 6.

It should be appreciated that the sampling function is known on both thesampling and the reconstruction sides of a system employing the TANStechnique. It should also be appreciated that the next sampling timedepends upon samples taken before that time. As shown in processingblock 46, at the appropriate time, the next sample is taken and thewindow (e.g. window 14 in FIG. 1) slides.

Decision box 50 implements a loop in which if a decision is made to keepsampling, then processing returns to processing block 44 where the nextsampling time is computed using the sampling function. If in decisionblock 44 a decision is made not to keep sampling, then processing ends.

Referring now to FIG. 4, another exemplary embodiment of atime-stampless adaptive nonuniform sampling process begins in processingblock 52 by obtaining a plurality of initialization samples. Processingthen proceeds to processing block 54 in which a sample is taken. Next,as shown in processing block 56, a sampling interval is computed using asampling function. Processing then proceeds to decision block 58 inwhich a loop Is implemented in which it is determined whether it is timeto sample (i.e. the process keeps checking to see when it is time tosample). Once it is time to sample, processing proceeds to processingblock 54 where a sample is taken and then processing flows to blocks 56,58.

Referring now to FIG. 5, another exemplary embodiment of a process fortime-stampless adaptive nonuniform sampling begins by obtaininginitialization samples and then predicting future samples as shown inprocessing blocks 60, 62.

Next, as shown in processing block 64, error values of the predictedsamples are computed and then as shown in processing block 66, the nextsampling time is determined based upon the error values.

Processing then proceeds to processing block 68 in which a next sampleis taken at an appropriate time. If the sample taken is a last sample ina given window, then the window “slides” as shown in processing block70.

Processing then proceeds to decision block 72 where a decision is madeas to whether additional samples should be taken. If in decisionblocking 72, a decision is made to keep sampling then processing flowsto processing block 74 in which future samples are predicted using thenew window. The processing in blocks 64-72 is then repeated.

If in decision block 72 a decision is made to not keep sampling thenprocessing ends.

Referring now to FIG. 6, a flow diagram for another exemplary embodimentof a time-stampless adaptive uniform sampling process begins byinitializing the system (e.g. by obtaining initialization samples) andpredicting future samples as shown in processing blocks 80, 82. Theprediction made in block 82 is then compared to a threshold value. Thethreshold value may correspond, for example, to an allowed distortionvalue. The threshold value may, of course, be based upon characteristicsother than distortion. For example, one may attempt to maintain acertain probability of keeping the approximation error within aspecified bound. Those skilled in the art can choose a characteristicappropriate for particular applications. In decision block 86, adecision is made as to whether a distortion threshold is exceeded. Ifthe distortion threshold is exceeded, then a sample is taken and thewindow slides as shown in processing blocks 88, 90. Processing thenreturns to processing block 80 and processing blocks 82-90 are repeated.

If in decision bock 86, a decision is made that the predicted values didnot exceed the threshold value, then processing flows to processingblock 92 where the sampling step is increased and processing returns toprocessing block 82 and processing blocks 82-86 (and optionally 88, 90)are repeated.

As described hereinabove, a general sampling framework which improves atrade-off between sampling rate and distortion by adapting to localcharacteristics of a signal being sampled has been described.Significantly, the sampling times are a function (e.g. a causalfunction) of previous samples, and thus there is no need to storedand/or communicate sampling times. Furthermore, the techniques describedherein above can be used with deterministic and stochastic modeling andanalysis.

To illustrate the concepts described herein above, next described arethree specific methods which can be used in accordance with the generalconcepts described herein for deterministic and stochastic signals. ATaylor series expansion technique and a Discrete-Time Valued techniqueare used for deterministic signals and a Generalized Linear Prediction(GLP) technique is used for stochastic signals.

In the examples to be described hereinbelow, the sampling scheme iscausal because the next sampling time depends upon samples taken beforethat time. In general, however, the sampling scheme can be designed tobe non-causal. It should also be noted that the reconstruction methodcan be causal or non-causal. The sampling scheme described herein is anadaptive process, because the sampling function depends upon localcharacteristics of the signal. Finding an appropriate sampling functionof the sampling scheme depends upon sampling requirements such as thesampling rate, the distortion requirement, etc. Table 1 illustrates ataxonomy of proposed sampling functions.

TABLE I Sampling Function Blind With Side Information DeterministicTaylor Series Expansion (TSE) Discrete Time Valued StochasticGeneralized Linear Prediction Linear Prediction with (GLP) SideInformation

The aim of the sampling scheme described is to balance between theaverage sampling rate and the reconstruction distortion.

It should be appreciated that this objective is different from theobjectives considered in change point analysis or active learning. Inthose techniques, the objective is to find points of the signal at whichthe statistical behaviors of the signal change, by causal or non-causalsampling, respectively.

Referring now to FIG. 7, a continuous signal 110 which may be expressedas X(t) has an i^(th) sample taken at time t_(i). For deterministicsignals, assuming the reconstruction method is line connection, thesampling function can be obtained via a Taylor Series Expansion (TSE).Thus, this method is referred to as the Taylor Series Expansion (TSE)method.

In this example, it is explained how a how a sampling function can becomputed for a specific set up. This method (as well as the so-calledGLP method to be described below) is referred to as a blind samplingmethod. Thus, Taylor's theorem is used to derive a suitable samplingfunction for deterministic signals. It should be appreciated thatvarious instantiations of TANS can be derived by using Taylor's theorem,depending on signals of interest, the allowed complexity of the samplingfunction f, and the distortion requirement.

In the example below, an explanation is provided as to how a samplingfunction can be computed for a specific setup where X(t) is a continuoussignal.

In this example, the following assumptions are made:

(C₁) the order of the sampling function is two. In other words, thesampling function is a function of three previously taken samples whichmay be expressed as: T_(i)=f(U_(j=i−m+1) ^(i−1){T_(j), Δ_(j)}).(C₂) The reconstruction method is connecting two adjacent samples by aline (this is referred to as non-causal linear interpolation).

Also, the distortion requirement is assumed to be as follows:

(C₃) If {circumflex over (X)}(t) is the reconstructed signal, then|X(t)−{circumflex over (X)}(t)|<D₁, for all t, where D1 is a samplingparameter.(C₄) it is assumed that the third derivative of X(t) is bounded by M;

The upper bound on the absolute value of the third derivative is used asa parameter in the sampling function and is also used in the analysis.

In the Taylor Series Expansion method of the locally adaptive samplingtechnique described herein, if |X′″(t)| is uniformly bounded by aconstant M, under assumptions (c1), (c2), the following samplingfunction satisfies the sampling requirement (c3).

T _(i)=arg max T

s.t. (C ₁ T+C ₂)T ² ≦D ₁  Equation (1)

where C₁ and C₂ are constants, defined as follows,

$C_{1} = \frac{M}{3}$$C_{2} = {\frac{\frac{{{\Delta \; i} - 1}}{{{Ti} - 1}} - \frac{{{\Delta \; i} - 2}}{{{Ti} - 2}}}{\frac{{Ti} - 1 + {Ti} - 2}{2}} + {\frac{M\;}{3}\frac{\left( {T_{i - 1} + T_{i - 2}} \right)^{2} + \left( T_{i - 1} \right)^{2}}{T_{i - 2}}}}$

It is insightful to investigate the behavior of T_(i), with respect todifferent involved parameters in Equation (1).

First, it should be noted that increasing D₁ leads to an increase inT_(i). Intuitively, the higher the allowed distortion, the lower thesampling rate, and the larger the sampling time intervals.

Also it should be noted that the first term of C2 (i.e.|Δ_(i−1)/T_(i−1)−Δ_(i−2)/T_(i−2)|/((T_(i−1)+T_(i−2))/2) can be viewed asan approximation of the second derivative of the signal X(t) (which maybe expressed as |X″(t)|) at time t_(i). It should be appreciated thatuse of a second derivative is but one of many ways to measure a signal.Since the reconstruction method is a first order linear filter, thehigher the second derivative, the faster changes of the signal.Therefore, by increasing

$\frac{\frac{{{\Delta \; i} - 1}}{{{Ti} - 1}} - \frac{{{\Delta \; i} - 2}}{{{Ti} - 2}}}{\frac{{Ti} - 1 + {Ti} - 2}{2}},T_{i}$

the value of T_(i), should decrease.

While the above has considered deterministic signals, since the proposedsampling method is blind, it renders this method a well-suited choicefor uncertain signals. For example, the method may be used when a signalis deterministic over stochastically varying time intervals.

Next described is a family of sampling functions where sampling timeintervals are discrete values. Thus, this technique is referred to as adiscrete time-valued method.

In this method, the sampling rate is being adapted based upon anestimation of local variations of the signal. This method can be appliedto both continuous and discrete signals. In this example, a continuoustime version of a signal is considered; however, its discrete timeversion can be also be derived.

If |X′″(t)| (i.e. third derivative of X(t)) is uniformly bounded by aconstant M, under assumptions C₁ and C₂ mentioned above, an estimate oflocal variation of the signal at time t_(i), by using previously takensamples, can be written as follows:

${w\left( t_{i} \right)}\overset{\bigtriangleup}{=}\frac{\frac{{{\Delta \; i} - 1}}{{{Ti} - 1}} - \frac{{{\Delta \; i} - 2}}{{{Ti} - 2}}}{\frac{{Ti} - 1 + {Ti} - 2}{2}}$

The error of this estimation can be bounded as follows:

${{{w\left( t_{i} \right)} -}}{X^{''}\left( t_{i} \right)}{{\leq {\frac{M}{3}\frac{\left( {T_{i - 1} + T_{i - 2}} \right)^{2} + \left( T_{i - 1} \right)^{2}}{T_{i - 2}}}}}$

Consider the following heuristic sampling function,

$T_{i} = \left\{ \begin{matrix}{f\; 1\left( {{Ti} - 1} \right)} & {{w({ti})} \leq {{th}\; 1\mspace{14mu} {and}\mspace{14mu} {Ti}} > {Tmin}} \\{{Ti} - 1} & {{{th}\; 1} \leq {w({ti})} < {{th}\; 2}} \\{f\; 2\left( {{Ti} - 1} \right)} & {{w({ti})} \geq {{th}\; 2\mspace{14mu} {and}\mspace{14mu} {Ti}} > {Tmax}}\end{matrix} \right.$

where

f ₁(T _(i−1))>T _(i−1) and

f ₂(T _(i−1))<T _(i−1).

-   -   T^(min) and T^(max) make sampling time intervals to be bounded.

Thresholds th₁ and th₂ depend upon signal characteristics and samplingrequirements. If W_((ti)) is smaller than a threshold, the signal'sslope variations are small and, since f₁(T_(i−1))>T_(i−1), the samplingrate can be decreased in this case. An analogous argument can beexpressed when w(t_(i)) is greater than a threshold.

An example of the above sampling function with linear increase ordecrease of T_(i) can be expressed as follows,

$T_{i} = \left\{ \begin{matrix}{T_{i - 1} + ɛ_{1}} & {{w\left( t_{i} \right)} \leq {{th}_{1}\mspace{14mu} {and}\mspace{14mu} T_{i}} > T^{\min}} \\T_{i - 1} & {{th}_{1} \leq {w\left( t_{i} \right)} < {th}_{2}} \\{T_{i - 1} - ɛ_{2}} & {{w\left( t_{i} \right)} \geq {{th}_{2}\mspace{14mu} {and}\mspace{14mu} T_{i}} < T^{\max}}\end{matrix} \right.$

where ∈₁ and ∈₂ are positive constants. Note that, given T_(i−1), thereare only three possibilities for T_(i), so, the sampling time intervalsare discrete.

Next described is a family of sampling functions based upon ageneralized linear prediction filter. Thus, this technique is referredto as a generalized linear prediction (GLP) method. This sampling methodis for discrete stochastic signals.

In this method, the signal is locally stationary. This family of signalsis described in detail below. The reconstruction method and thedistortion measure used are as follows:

-   (C7) the reconstruction method is using a generalized linear    prediction filter; and-   (C8) if {circumflex over (X)}[n] is the reconstructed signal, we    want to have E[(X[n]−{circumflex over (X)}[n])²]≦D₂, where D₂ is a    sampling parameter.

A generalized linear prediction filter for stationary signals is nextdescribed. Assume X[n] is a stationary signal and further assume thereare M samples of X[n] at times n−m₁, n−m₂, . . . , and n−m_(M). The aimis to predict linearly X[n] by using these known samples so that theexpected mean square error is minimized (MMSE predictor).

$\min\limits_{w_{m\; 1},\ldots \mspace{14mu},w_{mM}}\mspace{14mu} {E\left\lbrack {{\overset{\sim}{X}\lbrack n\rbrack}}^{2} \right\rbrack}$subject  to${\hat{X}\lbrack n\rbrack} = {\sum\limits_{k = 1}^{m}\; {w_{mk}{X\left\lbrack {n - m_{k}} \right\rbrack}}}$${\overset{\sim}{X}\lbrack n\rbrack} = {{X\lbrack n\rbrack} - {{\hat{X}\lbrack n\rbrack}.}}$

A solution of this linear optimization can be called w_(mi)* for 1≦I≦M.One difference of this setup with a traditional linear prediction is topredict X[n] by a set of non-uniform samples. In an optimal scheme, theerror term should be orthogonal to all known samples as expressed byEquation (2):

E[X[n−m _(k) ]{circumflex over (X)}*[n]]=0  Equation (2)

for k=1, . . . , M, where {circumflex over (X)}*[n]=X[n]−Σ_(k=1)^(M)w_(mk)*X[n−m_(k)].An auto-correlation function of X[n] can be written as shown in Equation(3):

r[i]=E[X[n]X ^(c) [n−i]]  Equation (3)

where X^(C)[n] is the conjugate of X[n]. Since this example deals withreal signals, without loss of generality conjugation effects areignored. Hence, by using Equations (2) and (3), the following set oflinear equations are found:

r[−mk]=Σ _(i=1) ^(M) w _(mi) *r[m _(i) −m _(k)]  Equation (4)

for k=1, . . . , M. This can be expressed in matrix form as shown inEquation (5):

m=[m ₁ , . . . , m _(M)]^(T)

X _(m) ^(n) =[X[n−m ₁ ], . . . , X[n−m _(M)]]^(T)

P=[r[−m ₁ ], . . . , r[−m _(M)]]^(T)

W* _(m) =[w* _(m1) , . . . , w* _(mM)]^(T)

R=E[(X _(m) ^(n))(X _(m) ^(n))^(T)].

Thus, linear equations of Equation (4) can be written as a matrixmultiplication

p=Rw _(mi)*

For X[n] with zero mean, define,

σ_(X) ² =E[|X[ _(n)]|² ]=r[0]

σ_(X) ² =E[|{tilde over (X)}*[ _(n)]|²|

Thus, for generalized liner prediction:

p=Rw _(mi)*

σ_(X) ² =r[0]−p ^(T) w _(m)*

For LAS with a generalized linear prediction filter, and assuming that asignal [n] is a locally stationary signal, locally stationary processescan be used as a tool to model systems where their statistical behaviorvaries with time. Definitions of a locally stationary process areprovided in S. Mallat, G. Papanicolaou, and Z. Zhang, “Adaptivecovariance estimation of locally stationary processes,” Annals ofStatistics, vol. 26, no. 1, pp. 1-47, 1998 and D. Donoho, S. Mallat, andR. von Sachs, “Estimating covariances of locally stationary processes:rates of convergence of best basis methods,” Statistics, StanfordUniversity, Stanford, Calif., USA, Tech. Rep, 1998.

Intuitively, a locally stationary process is a process where one canapproximate its local covariance coefficients within an error. Theabove-mentioned Mallat Reference approximates the covariance of alocally stationary process by a covariance which is diagonal in basis ofcosine packets, while the above-mentioned Donoho reference proposes amethod to estimate the covariance from sampled data.

For simplicity, a fixed window size is assumed. It should beappreciated, of course, that this can also vary by time.

A window W_(L)[n] with length L can be defined as follows,

${W_{L}\lbrack n\rbrack} = \left\{ \begin{matrix}1 & {0 \leq n \leq {L - 1}} \\0 & {otherwise}\end{matrix} \right.$

X_(L) ^(ni)[n] is a truncated version of X[n] which has its samples overn_(i)−L+1≦n≦n_(i).

That is:

X _(L) ^(ni) [n]=X[n],W _(L) [n−ni+L−1].

For X_(L) ^(ni)[n], auto-correlation coefficients can be written asfollows,

r ^(ni) _(l) [k]=E[X _(L) ^(ni) [n]X _(L) ^(ni) [n−k]].

By using these coefficients, for m=[0, l, . . . , M−l]^(T), where M<L,matrices can be defined as (X_(L) ^(ni))_(L), P_(L) ^(ni) ^(i) ,(w_(m)*)_(L) ^(ni) and R_(L) ^(ni) similarly to Equation (5) above.Since X[n] is assumed to be locally stationary, for any time n₀ and anygiven ∈, there exists an appropriate L such that,

|E[|{tilde over (X)}[n _(i)+1]|² ]−r ^(ni) _(L)[0]+(P ^(ni) _(L))T(w*_(m))^(ni) _(L)|<∈,

Where

{tilde over (X)}[n _(i)+1]|=X[n _(i)+1]−((w* _(m))^(ni) _(L))^(T)(X^(ni) _(m))_(L)

L is referred to as a window size of the locally stationary signal andL₀ and L₁ are defined as the minimum and the maximum allowed windowsizes, respectively. The stochastic nature of the signal affects L₀ andL₁.

Intuitively, since X[n] is locally stationary, its MMSE linearprediction filter with locally estimated autocorrelation coefficientsleads to an approximately same error as the one of stationary signals.Now, a setup of LAS by using a MMSE generalized linear prediction filterfor locally stationary signals can be introduced. Except being locallystationary, there are no other assumptions on the signal. Hence, thismethod is referred to as a blind sampling method.

Suppose X[n] is a locally stationary signal with window size L and thereare M samples X[n_(i)], X[n_(i−m2)] . . . X[n_(i−mM)]′ where 0<m₂<m₃< .. . <m_(M)<L. Now, consider a truncated signal X_(L) ^(ni)[n] as definedabove. If only taken samples of this truncated signal are used (i.e.X[n−mi] for 1≦i≦M), the approximations for R_(L) ^(ni), P_(L) ^(ni) and(w_(m)*)_(L) ^(ni).can be computed and are referred to as {circumflexover (R)}_(L) ^(ni), {circumflex over (P)}_(L) ^(ni) and (ŵ_(m)*)_(L)^(ni) respectively.

If L is sufficiently larger than L₀, we will have enough known samplesin the truncated signal and these approximations can be close to R_(L)^(ni), P_(L) ^(ni) and (w_(m)*)_(L) ^(ni). Then, one can linearlypredict X[n_(i)+N_(i)] by using samples X[n_(i)], X[n_(i−m2)]′ . . . ,and X[n_(i−mM)]. It is assumed that parameter L₁ of the locallystationary signal is sufficiently large.

Hence, by using σ_(X) ²=r[0]−p^(T)w_(m)* and |E[|{tilde over(X)}[n_(i)+1]|²]−r^(ni) _(L)[0]+(P^(ni) _(L))T(w*_(m))^(ni) _(L)|<∈,

|E[|{tilde over (X)}[ni+1]|² ]−{tilde over (r)} _(L)^(ni)[0]+({circumflex over (P)} _(L) ^(ni))T(ŵ _(m)*)_(L) ^(ni)|<∈

where ∈ is a small positive constant and m=[N_(i), N_(i+m2), . . . ,N_(i+mM)]. The reconstructed signal can be written as,

{circumflex over (X)}[n _(i) +n _(i)]=((ŵ _(m)*)_(L) ^(ni))^(T) X _(m)^(ni+Ni).

A sampling function for this scheme chooses the greatest possible N_(i)to have the expected error less than a threshold value D2. Thus, thissampling function can be written as the following linear optimizationsetup:

Max N _(i)

s.t. {circumflex over (X)}[n _(i) +N _(i)]=((ŵ _(m)*)_(L) ^(ni))^(T) X_(m) ^(ni+Ni).

{circumflex over (P)} _(L) ^(ni) ={circumflex over (R)} _(L) ^(ni)(ŵ_(m)*)_(L) ^(ni)

{circumflex over (r)} _(L) ^(ni)[0]−({circumflex over (P)} _(L)^(ni))^(T)(ŵ _(m)*)_(L) ^(ni) |<D ₂

It should be noted that, if the window size L is sufficiently largerthan the minimum allowed window size L₀, there are enough known samplesin the window, and these approximations would be appropriate. However,if there are not enough known samples in the window, autocorrelationcoefficients of the previous window can be used.

Next described is a sampling function based upon linear prediction withside information about the signal. Hence, this method is non-blind.Consider a signal model described as:

X[n]=α _(θn) X[n−1]+Z _(θn) [n]

where θ_(n) is the state of a hidden Markov chain (MC) with statetransition probability p.

Suppose its parameters (i.e., α_(θn)) are known for every n. Also,assume that the transition probability of the underlying MC, p, issmall. These parameters form the side information. For reconstruction, alinear prediction filter is used.

Consider A={(R, MSE)} as a set of achievable rate distortion pairs forthis signal model and consider the MSE of the error as a distortionmeasure. Similarly, A_(s) can be defined as A_(s)={(R_(s), MSE_(s))}, aset of achievable rate-distortion pairs within state s. The next sampleis taken when the prediction error (or, the noise variance) exceeds athreshold value D3.

For the signal model described above (i.e. X[n]=α_(θn)X[n−1]+Z_(θn)[n]),the following rate-distortion pairs are achievable,

$\begin{matrix}{A = {\left\{ {R,{MSE}} \right)\left( {R,{MSE}} \right)}} \\\left. {= {{\frac{1}{2}\left( {{1/K_{0}},{MSE}_{0}} \right)} + {\frac{1}{2}\left( {{1/K_{1}},{MSE}_{1}} \right)}}} \right\}\end{matrix}$${where},{{MSE}_{s} = {\sum\limits_{i = 1}^{{Ks} - 1}\; {\left( {1 - \alpha_{s}^{2\; i}} \right)/K_{s}}}}$and $K_{s} = \left\{ \begin{matrix}{{{\log \left( {1 - D_{3}} \right)}/2}\; \log \; \alpha_{s}} & {\alpha_{s} \notin \left\{ {0,1} \right\}} \\1 & {\alpha_{s} = 0} \\0 & {\alpha_{s} = 1}\end{matrix} \right.$

where 0≦D3≦1 is a sampling parameter and s∈{0, 1} represents the stateof the Markov Chain. By using X[n]=α_(θn)X[n−1]+Z_(θn)[n]) for eachstate s, results in:

$\begin{matrix}{{X\left\lbrack {n + K} \right\rbrack} = {{\alpha \; {{KsX}\lbrack n\rbrack}} + {\sum\limits_{i = 1}^{K}\; {{z\left\lbrack {n + i} \right\rbrack}a_{j}^{k - i}}}}} \\{= {{\alpha \; {{KsX}\lbrack n\rbrack}} + {Z_{s,K}.}}}\end{matrix}$ Hence, σ_(Zx, K)² = 1 − α_(s)^(2 K)

Consider a sample when the prediction error (or, the noise variance)exceeds a threshold value D3. Hence, for α_(s)∉{0, 1} a maximum value ofK is chosen to have:

1−α_(s) ^(2K) ≦D ₃

Hence, at state s,

K _(s)=log(1−D ₃)/2 log(α_(s))

when α_(s)∉{0, 1}.It should be noted that cases when α_(s)=0 or α_(s)=1 are trivial.Suppose MSE_(s)(l)=E[(X[n+l]−{circumflex over (X)}[n+l])²] at state s.If samples are taken at times n and n+K_(s), thenMSE_(s)(0)=MSE_(s)(K_(s))=0.For l∉{0, K_(s)}, MSE_(s)(l)=σ_(Zs,i) ²=1−σ_(s) ^(2l)

Hence, the average MSE at each state s (called MSE_(s)) is:

MSE_(s)=Σ_(i=1) ^(Ks−1)(1−α_(s) ^(2K))/K _(s)

The foregoing description has in some instances been directed tospecific embodiments. It will be apparent, however, that variations andmodifications may be made to the described embodiments, with theattainment of some or all of their advantages.

For instance, a family of locally adaptive sampling schemes (sometimereferred to as LAS) which can be applied on both stochastic signals hasbeen described. In this sampling family, time intervals between samplescan be computed by using a function of previously taken samples(referred to herein as a sampling function). Hence, although it is anon-uniform sampling, it is not necessary to keep sampling times. Theaim of this sampling scheme is to have the average sampling rate and thereconstruction error satisfy some requirements. Four different schemesof LAS have been described to help explain different properties: TaylorSeries Expansion (TSE) and Discrete Time Valued (DTV) methods fordeterministic signals; Generalized Linear Prediction (GLP) and LinearPrediction with Side Information (LPSI) methods for stochastic signals.TSE and GLP were referred to as blind methods since they included ageneral condition on the considered signal (bounded third derivative forTSE, and being locally stationary for GLP). However, DTV and LPSImethods are non-blind, because the sampling scheme is specificallydesigned for a known signal model.

Accordingly, in view of the description provided hereinabove, it shouldnow be apparent to those of ordinary skill in the art that the systemsand processes described herein find use in a wide variety ofapplications.

Sensing, for example, is a process performed in any system interactingwith the real world and one of the main power consumption contributorsof low power devices. For this reason, the framework described hereincan be applied to a wide variety of different sensing and other systems,enabling the development of new applications in the future.

For example, several research projects refer to continuous healthmonitoring by using either custom sensors or platforms based oncommercial mobile phones, recording ECG signals and other human's vitalinformation. The current approach for the design of these systems is tocontinuously sample the considered signals at a rate greater than orequal to the maximum rate of happening an event of interest. However, inthe approach described herein, the sampling rate is adapted by usingpreviously taken samples as described herein above. Furthermore, it isnot necessary to keep sampling times. Hence, the scheme described hereinachieves a lower power consumption in the sensing process.

For example, a generic sampling function can be designed by bounding thederivative of the signal by the integral of its Fourier transform andthen using the Cauchy-Schwarz inequality as follows:

T _(i)=[(p/c)(1/(w(t _(i))²)]^(1/3)

In which:

p is the signal power;

c is a sampling parameter;

w(t_(i)) is an approximation of a derivative of a signal at time t_(i)where

i.e., w(t_(i))=abs[(Δ_(i−1))/(T_(i−1)).

It should be noted that the higher the derivative at time t_(i), thehigher the signal variations at this time, and the smaller the samplingstep size. It should also be noted that, this sampling function is notdesigned specifically for ECG signals. In some simulations on ECGsignals, a sampling function designed by using characteristics of ECGsignals to provide more gain was used. Simulation results with real ECGsignals indicate advantages of the approach described herein, reducingby approximately an order of magnitude the sampling rate compared touniform sampling.

FIG. 8, for example, illustrates an ECG signal while FIGS. 8A and 8Billustrate two reconstructed ECG signals using the same number ofsamples (i.e., a down sampling factor of 10). FIG. 8A illustrates areconstructed ECG signal using the techniques described herein whileFIG. 8B illustrates a reconstructed ECG signal by uniform sampling. Onecan see that, the approach used in FIG. 8A provides a betterreconstructed signal given the same sampling rate.

The framework described herein can also be applied to a majority ofapplications that involve the interaction of smartphones and humans aswell as a surrounding environment. For instance, the use of smartphones,equipped with several sensors has been proposed for traffic monitoring,sound sensing, location tracking and many other scenarios. Although mostof these proposed ideas seem promising applications in the future, thestrict power constraints of portable devices are a main obstacle intheir realizations. It is believed that the framework described hereincan be an efficient solution in applications requiring continuoussampling and interaction with the real world. This is particularly truein view of data showing that sensors in a modern smartphone can consumeapproximately up to 60% of the overall power consumption when the phoneis in sleep mode.

While exemplary applications of TANS have been describe herein, those ofordinary skill in the art will appreciate that TANS can also be used ina wide variety of other applications.

Accordingly, it is submitted that that the concepts and techniquesdescribed herein should not be limited to the described embodiments butrather should be limited only by the spirit and scope of the appendedclaims.

What is claimed is:
 1. A method of sampling a signal, the method comprising: (a) obtaining an initial number of samples from the signal; (b) predicting a future sample based upon at least some of the initial number of samples; (c) computing an error value of the predicted future sample; and (d) comparing the error value to a threshold value.
 2. The method of claim 1 further comprising: (e) in response to the error value exceeding the threshold value, obtaining a next sample.
 3. The method of claim 2 further comprising: (f) in response to the error value not exceeding the threshold value, increasing a sampling time and computing a new predicted future sample.
 4. The method of claim 3 further comprising recovering sampling times using the sample values and sample times from a predetermined number of samples obtained from the signal.
 5. The method of claim 3 wherein obtaining an initial number of samples comprises measuring M samples of a signal.
 6. The method of claim 5 further comprising recovering sampling times using the sample values and sample times of the first M sample times obtained from the signal.
 7. The method of claim 1 wherein predicting a future sample based upon at least some of the initial number of samples comprises predicting a future sample based upon all of the initial number of samples.
 8. The method of claim 1 wherein the threshold value corresponds to an allowed distortion value.
 9. A sampling method comprising: (a) obtaining M samples of a signal; (b) computing a next sample time as a function of previously obtained samples of the signal wherein the sampling function is adaptive and time-stampless; and (c) recovering sampling times using values and sample times from the M samples obtained from the signal.
 10. The method of claim 9 wherein computing a next sample time as a function of previously obtained samples comprises computing a next sample time as a function of only M previously obtained samples.
 11. A method of sampling a signal, the method comprising: (a) obtaining M samples of the signal; and (b) computing a next sample time as a function of only the M previously obtained samples of the signal wherein the sampling is adaptive and self-synchronizing.
 12. The method of claim 11 further comprising: (c) recovering sampling times using the sample values and the sample times taken of the first M samples of the signal.
 13. A method for sampling a signal, the method comprising: (a) obtaining a first set of signal samples; (b) using the first set of signal samples to determine a sampling function to compute time intervals between samples; and (c) sampling the signal at a time determined by the sampling function.
 14. The method of claim 13 wherein the sampling function is based upon a set of the M most recently taken samples.
 15. The method of claim 13 wherein the sampling function is selected in accordance with sampling requirements and distortion requirements.
 16. The method of claim 13 wherein: in response to the signal being a stochastic signal, the sampling function is based upon one of: (1) the greedy method; or (2) a dynamic programming technique; and in response to the signal being a deterministic signal, the sampling function is based upon one of: (1) a Taylor expansion series; (2) a discrete-time valued function.
 17. A method comprising: (a) obtaining a first set of samples from a signal; (b) using the first set of samples to compute an interval using a sampling function wherein the sampling function determines the interval at which a next sample should be obtained; and (c) obtaining a next sample at the interval computed in (b).
 18. The method of claim 17 further comprising: (a) using a plurality of the previous samples including at least the last sample taken to compute a next interval at which a next sample should be taken; (b) repeating (d).
 19. The method of claim 18 further comprising providing to a reconstruction receiver, times at which each of the samples in the first set of samples were obtained.
 20. A method of adaptive, nonuniform sampling, the method comprising: identifying a sampling function; obtaining a first set of M samples from a signal wherein the first set of M samples includes sample times and sample values; obtaining additional samples from the signal wherein the samples are obtained only when the samples are innovative and wherein only sample values are obtained; providing to a receiver: (a) the sampling function; (b) the sample times of the first M samples; (c) the values of the additional samples; and recovering sampling times of the additional samples in the receiver using only the sampling function, the obtained sample values and sample times taken of the first M samples of the signal.
 21. The method of claim 20 wherein recovering sampling times in a receiver comprises computing the sampling times via the sampling function.
 22. The method of claim 20 wherein the signal being sampled corresponds to a stochastic signal and wherein the stochastic signal is locally stationary and wherein the sampling function is based upon a linear prediction filter which utilizes a General Linear Prediction (GLP) sampling function.
 23. The method of claim 20 wherein the signal being sampled corresponds to a stochastic signal having a known signal model and the sampling function is based upon a linear prediction filter which utilizes a linear prediction sampling function with side information (LPSI)
 24. A system comprising: (a) sampling system having an input and an output, the sampling system comprising: signal sampling means configured to receive an input signal from the input of the sampling system and configured to convert the input signal to a stream of digital bits; and control means, coupled to said signal sampling means, said control means for controlling the times at which said signal sampling means samples the input signal, said control means implementing a sampling function which controls said signal sample means to take samples only when the samples are innovative.
 25. The system of claim 24 further comprising a transmission system having an input coupled to the output of said sampling system and having an output, said transmission system configured to receive signals from said sampling system and to transmit the signals at the output thereof.
 26. The system of claim 25 further comprising a receiving system configured to receive signals provided by said transmission system, said receiving system comprising means for recovering sampling times from the received signals.
 27. The system of claim 24 wherein said signal sampling means is provided as an analog-to-digital converter.
 28. The system of claim 24 wherein said control means is provided as a digital signal processor.
 29. The system of claim 28 wherein said digital signal processor implements a real time compression scheme which adaptively compresses a signal using its local properties causally. 